The First SUSTech Weekend Workshop in Number Theory and Arithmetic Geometry

Title: Arithmetic purity of strong approximation

 

Speaker: Prof. Yang Cao, University of Science and Technology of China

 

Abstract: Strong approximation with Brauer-Manin obstruction is defined by Colliot-Thélène and Xu to study the local-global principle for the integral points. For an algebraic variety, inspired by analytic number theory, we consider the density of integral points with coprime values in adelic space: the arithmetic purity of strong approximation.

 

On the other side, for a semi-simple, simply connected k-simple linear algebraic group G, it is conjectured by Wittenberg that G satisfies arithmetic purity: the complement of any codimension >=2 closed subset satisfies strong approximation. We prove this conjecture for k-isotropic groups by an analogue of fibration method and  for Spin groups by using the density of rational points with almost prime polynomial values. This is joint work with Zhizhong Huang.

Title: Hilbert's Tenth Problem and Further Developments

 

Speaker: Prof. Zhi-Wei Sun, Nanjing University

 

Abstract: Hilbert's Tenth Problem (HTP) asked for an algorithm to determine wether an arbitrary polynomial equation with integer coefficients has solutions over the ring of integers. This was finally solved negatively by Yu. Matiyasevich in 1970, based on a 1961 paper by M. Davis, H. Putnam and J. Robinson.

 

In this three-hour talk, we introduce the theory of computability, the ingenious solution of HTP, and its further developments such as Matiyasevich's 9 unknowns theorem and the speaker's 11 unknowns theorem.

 

 

Title: Kloostermann refinement and large sieve inequality

 

Speaker: Prof. LiLu Zhao, Shandong University

 

Abstract: The circle method with Kloostermann refinement and the large sieve inequality are two important methods in analytic number theory. In this talk, we give a brief introduction to the above two methods. Some applications will be discussed.